Lemma 18.30.7. In Situation 18.30.5 assume (1) and (2) hold.

Every sheaf of sets is a filtered colimit of sheaves of the form

18.30.7.1\begin{equation} \label{sites-modules-equation-towards-constructible-sets} \text{Coequalizer}\left( \xymatrix{ \coprod \nolimits _{j = 1, \ldots , m} h_{V_ j}^\# \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{i = 1, \ldots , n} h_{U_ i}^\# } \right) \end{equation}with $U_ i$ and $V_ j$ in $\mathcal{B}$.

If $\mathcal{O}$ is a sheaf of rings, then every $\mathcal{O}$-module is a filtered colimit of sheaves of the form

18.30.7.2\begin{equation} \label{sites-modules-equation-towards-constructible} \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i} \right) \end{equation}with $U_ i$ and $V_ j$ in $\mathcal{B}$.

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